Forming a DE from F=ma

A particle of mass is projected with speed along a straight horizontal track. The first section of the track has length . On this section of the track the motion is resisted by a
constant force of magnitude , where is a positive constant. The particle does not come to rest on this first section of the track.

Show that the speed of the particle at the end of the first section of the track is given by:

Using the equation , I got the following equation.

If we let the speed of the particle =

, however in the solutions they just have , what's happened to the , I would have thought that the forces acting on the paticle would be and ?